Homogeneous Penalizers and Constraints in Convex Image Restoration

  • Recently convex optimization models were successfully applied for solving various problems in image analysis and restoration. In this paper, we are interested in relations between convex constrained optimization problems of the form \(min\{\Phi(x)\) subject to \(\Psi(x)\le\tau\}\) and their non-constrained, penalized counterparts \(min\{\Phi(x)+\lambda\Psi(x)\}\). We start with general considerations of the topic and provide a novel proof which ensures that a solution of the constrained problem with given \(\tau\) is also a solution of the on-constrained problem for a certain \(\lambda\). Then we deal with the special setting that \(\Psi\) is a semi-norm and \(\Phi=\phi(Hx)\), where \(H\) is a linear, not necessarily invertible operator and \(\phi\) is essentially smooth and strictly convex. In this case we can prove via the dual problems that there exists a bijective function which maps \(\tau\) from a certain interval to \(\lambda\) such that the solutions of the constrained problem coincide with those of the non-constrained problem if and only if \(\tau\) and \(\lambda\) are in the graph of this function. We illustrate the relation between \(\tau\) and \(\lambda\) by various problems arising in image processing. In particular, we demonstrate the performance of the constrained model in restoration tasks of images corrupted by Poisson noise and in inpainting models with constrained nuclear norm. Such models can be useful if we have a priori knowledge on the image rather than on the noise level.

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Metadaten
Verfasserangaben:René Ciak, Behrang Shafei, Gabriele Steidl
URN (Permalink):urn:nbn:de:hbz:386-kluedo-28669
Dokumentart:Preprint
Sprache der Veröffentlichung:Englisch
Veröffentlichungsdatum (online):02.02.2012
Jahr der Veröffentlichung:2012
Veröffentlichende Institution:Technische Universität Kaiserslautern
Datum der Publikation (Server):02.02.2012
Seitenzahl:28
Fachbereiche / Organisatorische Einheiten:Fachbereich Mathematik
DDC-Sachgruppen:5 Naturwissenschaften und Mathematik / 50 Naturwissenschaften / 500 Naturwissenschaften und Mathematik
Lizenz (Deutsch):Standard gemäß KLUEDO-Leitlinien vom 16.11.2011
Neuere Dokument-Versionen:urn:nbn:de:hbz:386-kluedo-33476

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